3.471 \(\int \frac{-10+x^2}{4+9 x^2+2 x^4} \, dx\)

Optimal. Leaf size=22 \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

ArcTan[x/2] - (3*ArcTan[Sqrt[2]*x])/Sqrt[2]

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Rubi [A]  time = 0.0106643, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1166, 203} \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

ArcTan[x/2] - (3*ArcTan[Sqrt[2]*x])/Sqrt[2]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{-10+x^2}{4+9 x^2+2 x^4} \, dx &=-\left (3 \int \frac{1}{1+2 x^2} \, dx\right )+4 \int \frac{1}{8+2 x^2} \, dx\\ &=\tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0124791, size = 22, normalized size = 1. \[ \tan ^{-1}\left (\frac{x}{2}\right )-\frac{3 \tan ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-10 + x^2)/(4 + 9*x^2 + 2*x^4),x]

[Out]

ArcTan[x/2] - (3*ArcTan[Sqrt[2]*x])/Sqrt[2]

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Maple [A]  time = 0.007, size = 17, normalized size = 0.8 \begin{align*} \arctan \left ({\frac{x}{2}} \right ) -{\frac{3\,\arctan \left ( x\sqrt{2} \right ) \sqrt{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-10)/(2*x^4+9*x^2+4),x)

[Out]

arctan(1/2*x)-3/2*arctan(x*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.64696, size = 22, normalized size = 1. \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="maxima")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

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Fricas [A]  time = 1.27442, size = 65, normalized size = 2.95 \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="fricas")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)

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Sympy [A]  time = 0.132663, size = 20, normalized size = 0.91 \begin{align*} \operatorname{atan}{\left (\frac{x}{2} \right )} - \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-10)/(2*x**4+9*x**2+4),x)

[Out]

atan(x/2) - 3*sqrt(2)*atan(sqrt(2)*x)/2

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Giac [A]  time = 1.1067, size = 22, normalized size = 1. \begin{align*} -\frac{3}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) + \arctan \left (\frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-10)/(2*x^4+9*x^2+4),x, algorithm="giac")

[Out]

-3/2*sqrt(2)*arctan(sqrt(2)*x) + arctan(1/2*x)